Transcript Document

Fractals and Applications in Landscape Ecology
Jiquan Chen
University of Toledo
Feb. 18, 2009
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How long is the costal line on earth?
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Would the pathways of a species through different patches
be similar? why?
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Can one use the knowledge of fractals to simulate different
landscape patterns and processes?
Fractals and Applications in Landscape Ecology
Jiquan Chen
University of Toledo
Feb. 18, 2009
•
How long is the costal line on earth?
•
Would the pathways of a species through different patches
be similar? why?
•
Can one use the knowledge of fractals to simulate different
landscape patterns and processes?
The Euclidean dimension of a point is zero, of a line segment is
one, a square is two, and of a cube is three. In general, the
fractal dimension is not an integer, but a fractional dimensional
(i.e., the origin of the term fractal by Mandelbrot 1967)
Sierpinski Carpet generated by fractals
So what is the dimension of the Sierpinski triangle? How do we find
the exponent in this case? For this, we need logarithms. Note that,
for the square, we have N2 self-similar pieces, each with
magnification factor N. So we can write:
http://math.bu.edu/DYSYS/chaos-game/node6.html
Self-similarity
One of the basic properties of fractal images is the notion of selfsimilarity. This idea is easy to explain using the Sierpinski triangle.
Note that S may be decomposed into 3 congruent figures, each of
which is exactly 1/2 the size of S! See Figure 7. That is to say, if we
magnify any of the 3 pieces of S shown in Figure 7 by a factor of 2,
we obtain an exact replica of S. That is, S consists of 3 self-similar
copies of itself, each with magnification factor 2.
Triadic Koch Island
ln( N n1 / N n )
D
ln(rn / rn1 )
1) r1=1/2, N1=2
2) R2=1/4, N2=4
D=0
http://mathworld.wolfram.com/Fractal.html
A geometric shape is created following the same rules or by the
same processes – inducing a self-similar structure
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Coastal lines
Stream networks
Number of peninsula along the Atlantic coast
Landscape structure
Movement of species
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Wiens et al. 1997, Oikos 78: 257-264
Vector-Based
D
2 *ln( Pij )
ln( Aij )
Raster-Based
D
2 *ln(0.25 * Pij )
ln( Aij )
Figure 11: The Sierpinski hexagon and pentagon
n mice start at the corners of a regular n-gon of unit side length, each heading towards its closest neighboring mouse in a counterclockwise